1. Field
The following description relates to subsurface exploration technology, and more particularly, to a technology of imaging the subsurface structure of a target area by using waveform inversion in a frequency domain, Laplace domain, or a Laplace-Fourier domain.
2. Description of the Related Art
Subsurface exploration is used to identify the subsurface structure and geological characteristics of a specified area and, particularly, to find useful resources (such as oil) buried underground. As the use of underground resources increases, subsurface exploration is widely being conducted not only in land but also in the sea. Subsurface exploration in land or sea plays a crucial role in discovering fossil fuels (e.g., natural gas and oil) which are important energy sources, understanding the subsurface structure of a desired area, and detecting underground water.
One of the most important geological and physical characteristics for accurate subsurface imaging is the seismic propagation velocity of the subsurface. To obtain the seismic propagation velocity of the subsurface of a desired area of land or sea, research is being conducted on a method of receiving and analyzing seismic waves reflected or refracted by the desired area. In this method, the seismic propagation velocity of the subsurface of a target area, that is, an image of the subsurface structure of the target area is obtained by artificially exploding a source wavelet into the target area and performing a predetermined operation using seismic data, i.e., data on a seismic wave reflected or refracted by the target area.
One way of obtaining the seismic propagation velocity of the subsurface by using seismic data is ‘waveform inversion.’ ‘Waveform inversion’ is a technique of estimating a subsurface velocity model using prestack seismic data. In waveform inversion, an initial model of an area of interest is constructed, and modeled data is obtained by modeling algorithm. Then, the initial model is iteratively updated using the modeled data and the measured data of an area of interest to reproduce a subsurface structure model close to the actual subsurface structure of the area. That is, data on the actual subsurface structure of the area is obtained by iteratively updating parameters, which represent physical characteristics of the subsurface structure of the area, until errors between modeled data obtained modeling using the subsurface parameters and measured data obtained through actual field exploration satisfy a predetermined condition.
Waveform inversion is one method of analyzing the subsurface structure of an area, which is one of the goals of geophysical exploration, and various mathematical methods are suggested. Representative mathematical methods for waveform inversion is an ‘iterative least-squares method’ with various objective functions such as a ‘logarithmic norm’ method. With the development of computers, simple waveform inversion can be achieved using personal computers.
Generally, a unique solution does not exist in waveform inversion. Thus, a method of obtaining an optimal solution by adding a particular condition is used in waveform inversion. In this case, it is optional whether to give weight to convergence or to obtain a more accurate solution from a given measured data. Since an inverse model must be simplified and often requires extreme assumptions, it is important in waveform inversion to make the most of prior geological information related to geophysical characteristics.
If low-frequency data can be used and if there are no computational limitations, seismic waveform inversion has the advantage of providing a more detailed subsurface velocity model than travel time tomography or conventional velocity analysis. About thirty years ago, Lailly and others tackled the seismic inverse problem by using reverse time migration (Lailly, P. 1983, the Seismic Inverse Problem as a Sequence of Before Stack Migrations: Society for Industrial and Applied Mathematics, Philadelphia). Since then, geoscientists and applied mathematicians have used a similar back-propagation technique for waveform inversion.
However, real seismic data obtained through actual field exploration presents many obstacles to waveform inversion: the absence of low-frequency data, two-dimensional (2D) acoustic approximations of three-dimensional (3D) real earth wave propagation, and source-receiver coupling to the earth. Noise is perhaps the most important of these obstacles because ambient background vibrations always contaminate real seismic data.
Due to these obstacles, a gradient vector of an objective function for an arbitrary parameter pk may significantly vary for each frequency, each Laplace damping constant, or each Laplace-Fourier damping constant in waveform inversion in the frequency domain, the Laplace domain, or the Laplace-Fourier domain. In this case, each frequency, each Laplace damping constant, or each Laplace-Fourier damping constant may make a greatly different contribution when a gradient direction for all frequencies, all Laplace damping constants, or all Laplace-Fourier damping constants is determined by adding a gradient vector for each frequency, each Laplace damping constant, or each Laplace-Fourier damping constant. Furthermore, if iterations are performed by applying a gradient vector for all frequencies, all Laplace damping constants, or all Laplace-Fourier damping constants as it is, an additional operation such as a line search is required to determine a step length for parameter update.